12th Mathematics (Arts & Science) Part 2 Chapter 8 Solution (Digest) Maharashtra state board

Chapter 8 Binomial Distribution

Open with Full Screen in HD Quality

HD PDF

Project on Binomial Distribution

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes: success or failure. It is named after the Swiss mathematician Jacob Bernoulli.

Key Concepts:

1. Parameters:

   • $n$: The number of trials or experiments.
   • $p$: The probability of success in each trial.
   • $q = 1 - p$: The probability of failure in each trial.

2. Probability Mass Function (PMF): The probability mass function of the binomial distribution gives the probability of obtaining exactly $k$ successes in $n$ trials: $P(X = k) = \binom{n}{k} p^k q^{n-k}$ where $\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose $k$ successes out of $n$ trials.

3. Mean and Variance:

   • Mean (Expected Value): $\mu = np$
   • Variance: $\sigma^2 = npq$

4. Properties:

   • The binomial distribution is symmetric if $p = q = 0.5$.
   • As $n$ increases, the binomial distribution becomes increasingly bell-shaped and approaches a normal distribution.

5. Applications:

   • Coin Flipping: Modeling the number of heads in a series of coin flips.
   • Manufacturing: Predicting the number of defective products in a batch.
   • Biological Studies: Analyzing genetic crosses and inheritance patterns.
   • Survey Sampling: Estimating the proportion of a population with a certain characteristic.
   • Quality Control: Monitoring the success rate of a manufacturing process.

Example:

Suppose we have a biased coin with a probability of heads $p = 0.6$. We flip the coin 5 times. What is the probability of getting exactly 3 heads?

Using the binomial distribution formula: $P(X = 3) = \binom{5}{3} (0.6)^3 (0.4)^2$

$P(X = 3) = \frac{5!}{3!(5-3)!} (0.6)^3 (0.4)^2 = 10 \times 0.6^3 \times 0.4^2 = 0.2304$

So, the probability of getting exactly 3 heads in 5 coin flips with a biased coin where $p = 0.6$ is 0.2304.

The binomial distribution is a fundamental concept in probability theory and has widespread applications in various fields, including statistics, economics, and biology.